High precision differential abundance measurements in globular clusters: chemical inhomogeneities in NGC 6752

MNRAS 434, 3542 3565 (2013) Advance Access publication 2013 August 7 doi:10.1093/mnras/stt1276 High precision differential abundance measurements in globular clusters: chemical inhomogeneities in NGC 6752 David Yong, 1 Jorge Meléndez, 2 Frank Grundahl, 3 Ian U. Roederer, 4 John E. Norris, 1 A. P. Milone, 1 A. F. Marino, 1 P. Coelho, 5 Barbara E. McArthur, 6 K. Lind, 7 R. Collet 1 and Martin Asplund 1 1 Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia 2 Departamento de Astronomia do IAG/USP, Universidade de Sao Paulo, Rua do Matao 1226, Sao Paulo 05508-900, SP, Brasil 3 Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark 4 Carnegie Observatories, 813 Santa Barbara Street, Pasadena, CA 91101, USA 5 Núcleo de Astrofísica Teórica, Universidade Cruzeiro do Sul, R. Galvão Bueno 868, Liberdade 01506-000, São Paulo, Brazil 6 McDonald Observatory, University of Texas, Austin, TX 78712, USA 7 University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK Accepted 2013 July 10. Received 2013 July 9; in original form 2013 April 10 ABSTRACT We report on a strictly differential line-by-line analysis of high-quality UVES spectra of bright giants in the metal-poor globular cluster NGC 6752. We achieved high precision differential chemical abundance measurements for Fe, Na, Si, Ca, Ti, Cr, Ni, Zn, Y, Zr, Ba, La, Ce, Pr, Nd, Sm, Eu and Dy with uncertainties as low as 0.01 dex ( 2 per cent). We obtained the following main results. (1) The observed abundance dispersions are a factor of 2 larger than the average measurement uncertainty. (2) There are positive correlations, of high statistical significance, between all elements and Na. (3) For any pair of elements, there are positive correlations of high statistical significance, although the amplitudes of the abundance variations are small. Removing abundance trends with effective temperature and/or using a different pair of reference stars does not alter these results. These abundance variations and correlations may reflect a combination of (a) He abundance variations and (b) inhomogeneous chemical evolution in the pre- or protocluster environment. Regarding the former, the current constraints on Y from photometry likely preclude He as being the sole explanation. Regarding the latter, the nucleosynthetic source(s) must have synthesized Na, α, Fe-peak and neutron-capture elements and in constant amounts for species heavier than Si; no individual object can achieve such nucleosynthesis. We speculate that other, if not all, globular clusters may exhibit comparable abundance variations and correlations to NGC 6752 if subjected to a similarly precise analysis. Key words: stars: abundances Galaxy: abundances globular clusters: individual: NGC 6752. Downloaded from http://mnras.oxfordjournals.org/ at The Australian National University on November 7, 2013 1 INTRODUCTION Understanding the origin of the star-to-star abundance variations of the light elements in globular clusters is one of the major challenges confronting stellar evolution, stellar nucleosynthesis and chemical evolution. Arguably the first evidence for chemical abundance inhomogeneity in a globular cluster was the discovery of a CN strong star in M13 by Popper (1947). A large number of subsequent studies have confirmed the star-to-star variation in the strength of the Based on observations collected at the European Southern Observatory, Chile (ESO Programmes 67.D-0145 and 65.L-0165A). E-mail: yong@mso.anu.edu.au CN molecular bands in a given globular cluster, and these results have been extended to star-to-star abundance variations for the light elements Li, C, N, O, F, Na, Mg and Al (e.g. see reviews by Smith 1987; Kraft 1994; Gratton, Sneden & Carretta 2004; Gratton, Carretta & Bragaglia 2012). In light of the discovery of abundance variations in unevolved stars (e.g. Cannon et al. 1998; Gratton et al. 2001; Ramírez & Cohen 2002, 2003), the consensus view is that these light element abundance variations are attributed to a protocluster environment in which the gas was of an inhomogeneous composition. The interstellar medium from which some of the stars formed included material processed through hydrogen burning at high temperatures. The source of that material and the nature of the nucleosynthesis, however, remain highly contentious with intermediate-mass asymptotic giant branch (AGB) C 2013 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society

Precision abundance measurements in NGC 6752 3543 stars, fast rotating massive stars (FRMS) and massive binaries being the leading candidates (e.g. Fenner et al. 2004; Ventura & D Antona 2005; Decressin et al. 2007; de Mink et al. 2009; Marcolini et al. 2009). Recent discoveries of complex structure in colour magnitude diagrams reveal that most, if not all, globular clusters host multiple populations; the evidence consists of multiple main sequences, subgiant branches, red giant branches (RGBs) and/or horizontal branches (HBs) in Galactic (e.g. see Piotto 2009 for a review) and also extragalactic globular clusters (e.g. Mackey & Broby Nielsen 2007; Milone et al. 2009). When using appropriate photometric filters, all globular clusters show well-defined sequences with distinct chemical abundance patterns (Milone et al. 2012). These multiple populations can be best explained by different ages and/or chemical compositions. The sequence of events leading to the formation of multiple population globular clusters is not well understood (e.g. D Ercole et al. 2008; Bekki 2011; Conroy & Spergel 2011). Although the census and characterization of the Galactic globular clusters remains incomplete, they may be placed into three general categories: 1 (i) those that exhibit only light element abundance variations, which include NGC 6397, NGC 6752 and 47 Tuc (e.g. Gratton et al. 2001; Yong et al. 2005; D Orazi et al. 2010; Lind et al. 2011a; Campbell et al. 2013), (ii) those that exhibit light element abundance variations and neutron-capture element abundance dispersions such as M15 (e.g. Sneden et al. 1997, 2000; Sobeck et al. 2011) and (iii) those that exhibit light element abundance variations as well as significant abundance dispersions for Fe-peak elements 2 such as ω Cen, M22, M54, NGC 1851, NGC 3201 and Terzan 5 (e.g. Norris & Da Costa 1995; Yong & Grundahl 2008; Marino et al. 2009, 2011; Carretta et al. 2010, 2011; Johnson & Pilachowski 2010; Villanova, Geisler & Piotto 2010; Origlia et al. 2011; Roederer, Marino & Sneden 2011; Alves-Brito et al. 2012; Simmerer et al. 2013). At this stage, we do not attempt to classify a particularly unusual system like NGC 2419 (Cohen et al. 2010; Cohen, Huang & Kirby 2011; Cohen & Kirby 2012; Mucciarelli et al. 2012). Given the surprisingly large star-to-star variations in element abundance ratios in a given cluster, how chemically homogeneous are the well-behaved elements in the normal globular clusters (i.e. clusters in category (i) above)? The answer to this question has important consequences for testing model predictions, setting constraints on the polluters and understanding the origin and evolution of globular clusters. Sneden (2005) considered the issue of cluster abundance accuracy limits and selected the [Ni/Fe] ratio as an example. This pair of elements was chosen as they present numerous spectral lines in the uncomplicated yellow red region of the spectrum and share common nucleosynthetic origins in supernovae. Sneden (2005) noted that the dispersion in the [Ni/Fe] ratio in a cluster was 0.06 dex and appeared to show little apparent trend as a function of the number of stars observed in a survey or of year of publication. There are two possible reasons for the apparent limit in the σ [Ni/Fe] ratio. Perhaps clusters possess a single [Ni/Fe] ratio and the dispersion reflects the measurement uncertainties. Alternatively, globular clusters are chemically homogeneous in the [Ni/Fe] ratio at the 0.06 dex level. Bearing in mind this apparent limit in the [Ni/Fe] dispersion, in order to answer the question posed above, we require the highest possible precision when measuring chemical abundances. A number of recent studies have achieved precision in chemical abundance measurements as low as 0.01 dex (e.g. Meléndez et al. 2009, 2012; Alves-Brito et al. 2010; Nissen & Schuster 2010, 2011; Ramírez et al. 2010; Ramírez, Meléndez & Chanamé 2012). These results were obtained by using (i) high-quality spectra (R 60 000 and signal-to-noise ratios S/N 200 per pixel), (ii) a strictly differential line-by-line analysis and (iii) a well-chosen sample of stars covering a small range in stellar parameters (effective temperature, surface gravity and metallicity). Application of similar analysis techniques to high-quality spectra of stars in globular clusters offers the hope that high precision chemical abundance measurements (at the 0.01 dex level) can also be obtained. To our knowledge, the highest precision chemical abundance measurements in globular clusters to date at the 0.04 dex level include those of Yong et al. (2005), Gratton et al. (2005), Carretta et al. (2009b) and Meléndez & Cohen (2009). The aim of this paper is to achieve high precision abundance measurements in the globular cluster NGC 6752 and to use these data to study the chemical enrichment history of this cluster. 2 OBSERVATIONS AND ANALYSIS 2.1 Target selection and spectroscopic observations The targets for this study were taken from the uvby photometry by Grundahl et al. (1999). The sample consists of 17 stars located near the tip of the RGB (hereafter RGB tip stars) and 21 stars located at the bump in the luminosity function along the RGB (hereafter RGB bump stars). The list of targets can be found in Table 1. Observations were performed using the Ultraviolet and Visual Echelle Spectrograph (UVES; Dekker et al. 2000) on the 8.2 m Kueyen (VLT/UT2) telescope at Cerro Paranal, Chile. The RGB tip stars were observed at a resolving power of R = 110 000 and S/N 150 per pixel near 5140 Å while the RGB bump stars were observed at R = 60 000 and S/N 100 per pixel near 5140 Å. Analyses of these spectra have been reported in Grundahl et al. (2002) and Yong et al. (2003, 2005, 2008). The location of the program stars in a colour magnitude diagram can be found in fig. 1 in Yong et al. (2003). Based on multiband Hubble Space Telescope (HST) and groundbased Strömgren photometry, Milone et al. (2013) have identified three populations on the main sequence, subgiant branch and RGB of NGC 6752. These populations, which we refer to as a, b and c, exhibit distinct chemical abundance patterns: population a has a chemical composition similar to that of field halo stars (e.g. high O and low Na); population c is enhanced in N, Na and He ( Y 0.03) and depleted C and O; population b has a chemical composition intermediate between populations a and c with slightly enhanced He ( Y 0.01). Using the data from Milone et al. (2013), we can classify all program stars according to their populations. In the relevant figures, stars of populations a, b and c are coloured green, magenta and blue, respectively. 1 There are subtle, and not so subtle, differences within a given category. 2 Saviane et al. (2012) have identified a metallicity dispersion in NGC 5824. To our knowledge, there are no published studies of the light element abundances based on high-resolution spectroscopy, so we cannot yet place this globular cluster in category (iii). 2.2 Line list and equivalent width measurements The first step in our analysis was to measure equivalent widths (EWs) for a large set of lines. The line list was taken primarily from Gratton et al. (2003) and supplemented with laboratory measurements for Fe I from the Oxford group (Blackwell et al. 1979a;

3544 D. Yong et al. Table 1. Program stars and stellar parameters as defined in Section 2.3. Name1 a Name2 RA 2000 Dec. 2000 V T b eff log g b ξ b t [Fe/H] b (K) (cm s 2 ) (kms 1 ) (1) (2) (3) (4) (5) (6) (7) (8) (9) PD1 NGC 6752-mg0 19:10:58 59:58:07 10.70 3928 0.26 2.20 1.67 B1630 NGC 6752-mg1 19:11:11 59:59:51 10.73 3900 0.24 2.25 1.70 B3589 NGC 6752-mg2 19:10:32 59:57:01 10.94 3894 0.33 2.07 1.66 B1416 NGC 6752-mg3 19:11:17 60:03:10 10.99 4050 0.50 1.88 1.66 NGC 6752-mg4 19:10:43 59:59:54 11.02 4065 0.53 1.86 1.65 PD2 NGC 6752-mg5 19:10:49 59:59:34 11.03 4100 0.56 1.90 1.65 B2113 NGC 6752-mg6 19:11:03 60:01:43 11.22 4154 0.68 1.85 1.62 NGC 6752-mg8 19:10:38 60:04:10 11.47 4250 0.80 1.71 1.69 B3169 NGC 6752-mg9 19:10:40 59:58:14 11.52 4288 0.91 1.72 1.66 B2575 NGC 6752-mg10 19:10:54 59:57:14 11.54 4264 0.90 1.66 1.67 NGC 6752-mg12 19:10:58 59:57:04 11.59 4286 0.94 1.73 1.68 B2196 NGC 6752-mg15 19:11:01 59:57:18 11.68 4354 1.02 1.74 1.64 B1518 NGC 6752-mg18 19:11:15 60:00:29 11.83 4398 1.11 1.68 1.64 B3805 NGC 6752-mg21 19:10:28 59:59:49 11.99 4429 1.20 1.68 1.65 B2580 NGC 6752-mg22 19:10:54 60:02:05 11.99 4436 1.20 1.71 1.65 B1285 NGC 6752-mg24 19:11:19 60:00:31 12.15 4511 1.31 1.69 1.67 B2892 NGC 6752-mg25 19:10:46 59:56:22 12.23 4489 1.33 1.70 1.67 NGC 6752 0 19:11:03 59:59:32 13.03 4699 1.83 1.43 1.66 B2882 NGC 6752 1 19:10:47 60:00:43 13.27 4749 1.95 1.37 1.63 B1635 NGC 6752 2 19:11:11 60:00:17 13.30 4779 1.98 1.37 1.63 B2271 NGC 6752 3 19:11:00 59:56:40 13.41 4796 2.03 1.38 1.69 B611 NGC 6752 4 19:11:33 60:00:02 13.42 4806 2.04 1.38 1.65 B3490 NGC 6752 6 19:10:34 59:59:55 13.47 4804 2.06 1.33 1.64 B2438 NGC 6752 7 19:10:57 60:00:41 13.53 4829 2.10 1.32 1.86 c B3103 NGC 6752 8 19:10:45 59:58:18 13.56 4910 2.15 1.33 1.69 B3880 NGC 6752 9 19:10:26 59:59:05 13.57 4824 2.11 1.41 1.70 B1330 NGC 6752 10 19:11:18 59:59:42 13.60 4836 2.13 1.37 1.65 B2728 NGC 6752 11 19:10:50 60:02:25 13.62 4829 2.13 1.34 1.68 B4216 NGC 6752 12 19:10:20 60:00:30 13.64 4841 2.15 1.35 1.66 B2782 NGC 6752 15 19:10:49 60:01:55 13.73 4850 2.19 1.36 1.63 B4446 NGC 6752 16 19:10:15 59:59:14 13.78 4906 2.24 1.33 1.63 B1113 NGC 6752 19 19:11:23 59:59:40 13.96 4928 2.32 1.33 1.68 NGC 6752 20 19:10:36 59:56:08 13.98 4929 2.33 1.32 1.63 NGC 6752 21 19:11:13 60:02:30 14.02 4904 2.33 1.31 1.67 B1668 NGC 6752 23 19:11:12 59:58:29 14.06 4916 2.35 1.25 1.66 NGC 6752 24 19:10:44 59:59:41 14.06 4948 2.37 1.16 1.71 NGC 6752 29 19:10:17 60:01:00 14.18 4950 2.42 1.31 1.69 NGC 6752 30 19:10:39 59:59:47 14.19 4943 2.42 1.26 1.64 a PD1 and PD2 are from Penny & Dickens (1986) and BXXXX names are from Buonanno et al. (1986). b These stellar parameters are for the so-called reference star values (see Section 2.3 for details). c We exclude this star from the subsequent differential analysis due to its discrepant metallicity. Blackwell, Petford & Shallis 1979b; Blackwell et al. 1980, 1986; Blackwell, Lynas-Gray & Smith 1995), laboratory measurements forfeii from Biemont et al. (1991) and for various elements, the values taken from the references listed in Yong et al. (2005) (which are also listed in Tables 2 and 3). We used the DAOSPEC (Stetson & Pancino 2008) software package to measure EWs in our program stars. For the subset of lines we had previously measured using routines in IRAF, 3 we compared those values with the DAOSPEC measurements and found excellent agreement between the two sets of EW measurements for lines having strengths less than 100 må (see Fig. 1). For the 1542 lines with EW < 100 må, we find a mean difference EW(DY) EW(DAOSPEC) = 1.14 ± 0.05 må (σ = 1.92 må). For 3 Image Reduction and Analysis Facility (IRAF) is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with the National Science Foundation. our analysis, we adopted only lines with 5 må < EW < 100 må as measured by DAOSPEC. A further requirement was that a given line must be measured in every RGB tip star or every RGB bump star. That is, the line list for the RGB tip sample was different from the line list for the RGB bump sample, but for either sample of stars, each line was measured in every star within a particular sample. Due to the lower quality spectra for the RGB bump sample, we required lines to have EW 10 må. The line list and EW measurements for the RGB tip sample and for the RGB bump sample are presented in Tables 2 and 3, respectively. 2.3 Establishing parameters for reference stars In order to conduct the line-by-line strictly differential analysis, we needed to adopt a reference star. The reference star parameters were determined in the following manner. Note that since we did not know which reference stars would be adopted, the procedure was

Precision abundance measurements in NGC 6752 3545 Table 2. Line list for the RGB tip stars. Wavelength Species a L.E.P log gf mg0 b mg1 mg2 mg3 mg4 Source c Å (ev) (må) (må) (må) (må) (må) (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 6154.23 11.0 2.10 1.56 48.2 32.2 23.9 18.5 20.3 A 6160.75 11.0 2.10 1.26 74.7 53.1 42.1 34.2 37.7 A 5645.61 14.0 4.93 2.14 16.0 16.3 15.8 15.6 15.9 A 5665.56 14.0 4.92 2.04 20.3 20.4 20.4 19.4 19.4 B 5684.49 14.0 4.95 1.65 35.0 36.1 34.2 34.1 33.3 B a The digits to the left of the decimal point are the atomic number. The digit to the right of the decimal point is the ionization state ( 0 = neutral, 1 = singly ionized). b Star names are abbreviated. See Table 1 for the full names. c A = log gf values taken from Yong et al. (2005) where the references include Den Hartog et al. (2003), Ivans et al. (2001), Kurucz & Bell (1995), Prochaska et al. (2000) and Ramírez & Cohen (2002); B = Gratton et al. (2003); C = Oxford group including Blackwell et al. (1979a,b, 1980, 1986, 1995); D = Biemont et al. (1991). This table is published in its entirety in the electronic edition of the paper. A portion is shown here for guidance regarding its form and content. Table 3. Line list for the RGB bump stars. Wavelength Species a L.E.P log gf 0 b 1 2 3 4 Source c Å ev må må må må må (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) 5682.65 11.0 2.10 0.71 52.1 18.6 56.1 15.3 50.2 A 5688.22 11.0 2.10 0.40 77.0 31.9 75.5 27.3 73.8 A 5684.49 14.0 4.95 1.65 24.4 22.9 22.5 20.8 23.6 B 5708.40 14.0 4.95 1.47 38.3 28.7 33.9 28.4 30.4 B 5948.55 14.0 5.08 1.23 43.5 36.9 39.4 31.8 37.5 A a The digits to the left of the decimal point are the atomic number. The digit to the right of the decimal point is the ionization state ( 0 = neutral, 1 = singly ionized). b Star names are abbreviated. See Table 1 for the full names. c A = log gf values taken from Yong et al. (2005) where the references include Den Hartog et al. (2003), Ivans et al. (2001), Kurucz & Bell (1995), Prochaska et al. (2000) and Ramírez & Cohen (2002); B = Gratton et al. (2003); C = Oxford group including Blackwell et al. (1979a,b, 1980, 1986, 1995); D = Biemont et al. (1991). This table is published in its entirety in the electronic edition of the paper. A portion is shown here for guidance regarding its form and content. applied to all stars. Following our previous analyses of these spectra, effective temperatures, T eff, were derived from the Grundahl et al. (1999) uvby photometry using the Alonso, Arribas & Martínez- Roger (1999) T eff :colour:[fe/h] relations. Surface gravities, log g, were estimated using T eff and the stellar luminosity. The latter value was determined by assuming a mass of 0.84 M, a reddening E(B V) = 0.04 (Harris 1996) and bolometric corrections taken from a 14 Gyr isochrone with [Fe/H] = 1.54 from VandenBerg et al. (2000). The model atmospheres used in the analysis were the one dimensional, plane parallel, local thermodynamic equilibrium (LTE), α enhanced, [α/fe] =+0.4, NEWODF grid of ATLAS9 models by Castelli & Kurucz (2003). We used linear interpolation software (written by Dr Carlos Allende Prieto and tested in Allende Prieto et al. 2004) to produce a particular model. (See Mészáros & Allende Prieto 2013 for a discussion of interpolation of model atmospheres.) Using the 2011 version of the stellar line analysis program MOOG (Sneden 1973; Sobeck et al. 2011), we computed the abundance for a given line. The microturbulent velocity, ξ t, was set, in the usual way, by forcing the abundances from Fe I lines to have zero slope against the reduced equivalent width, EW r = log (W λ /λ). The metallicity was inferred from Fe I lines. We iterated this process until the inferred metallicity matched the value adopted to generate the model atmosphere (this process usually converged within three iterations). (We exclude the RGB bump star NGC 6752 7 (B2438) due to its discrepant iron abundance, most likely resulting from a photometric blend which affected the T eff and log g values.) 2.4 Line-by-line strictly differential stellar parameters Following Meléndez et al. (2012), we determined the stellar parameters using a strictly differential line-by-line analysis between the program stars and a reference star. Given the difference in T eff between the RGB tip and RGB bump samples, we treated each sample separately. For the RGB tip stars, we selected NGC 6752-mg9 to be the reference star since it had a T eff value close to the median for the RGB tip stars and the O/Na/Mg/Al abundances were also close to the median values. These decisions were motivated by the expectation that the errors in the derived stellar parameters, and therefore errors in the chemical abundances, would increase if there was a large difference in T eff between the program star and the reference star. Thus, we selected a star with T eff close to the median value to minimize the difference in T eff between the program stars and the reference star. Similarly, we were concerned that large differences in the abundances of O/Na/Mg/Al between the program star and the

3546 D. Yong et al. Figure 1. Comparison of EWs measured using IRAF (DY) and DAOSPEC.The upper panel shows all lines (N = 1795). The lower panel shows the distribution of the EW differences for the 1542 lines with EW DY < 100 må (i.e. measured using IRAF). We superimpose the Gaussian fit to the distribution and write the relevant parameters associated with the fit as well as the mean and dispersion. reference star could increase the errors in the derived stellar parameters and chemical abundances. Again, selecting the reference star to have O/Na/Mg/Al abundances close to the median value minimizes the abundance differences between the program stars and the reference star. Application of a similar approach to the RGB bump sample resulted in the selection of NGC 6752 11 as the reference star. To determine the stellar parameters for a program star, we generated a model atmosphere with a particular combination of effective temperature (T eff ), surface gravity (log g), microturbulent velocity (ξ t ) and metallicity, [m/h]. The initial guesses for these parameters came from the values in Section 2.3. Using MOOG, we computed the abundances for Fe I and Fe II lines. We then examined the line-by-line Fe abundance differences. Adopting the notation from Meléndez et al. (2012), the abundance difference (program star reference star) for a line is program star δa i = Ai reference star Ai. (1) We examined the abundance differences for Fe I as a function of lower excitation potential. We forced excitation equilibrium by imposing the following constraint ( ) δa Fe i I = 0. (2) (χ exc ) Next, we considered the abundance differences for Fe I as a function of reduced equivalent width, EW r, and imposed the following constraint ( ) δa Fe i I = 0. (3) (EW r ) For any species, Fe I in this example, we then defined the average abundance difference as Fe I = δa Fe 1 i I = N N i=1 δa Fe I i. (4) Similarly, we defined the average Fe II abundance as Fe II Fe II = δai, and the relative ionization equilibrium as Fe I Fe II = Fe I Fe II = δa Fe i I δa Fe II i = 0. (5) Unlike Meléndez et al. (2012), we did not take into account the relative ionization equilibria for Cr and Ti, nor did we consider nonlocal thermodynamic equilibrium (NLTE) effects for any species. We note that while departures from LTE are expected for Fe I for metal-poor giants (Lind, Bergemann & Asplund 2012), the relative NLTE effects across our range of stellar parameters are vanishingly small. The final stellar parameters for a program star were obtained when equations (2), (3) and (5) were simultaneously satisfied and the derived metallicity was identical to that used in generating the model atmosphere. Regarding the latter criterion, we provide the following example. The metallicity of the reference star NGC 6752- mg9 was [Fe/H] = 1.66 when adopting the Asplund et al. (2009) solar abundances and the photometric stellar parameters described in Section 2.3 (see Table 1). For star NGC 6752-mg8, the average abundance difference for Fe I, and also Fe II given equation (5), was δa Fe i I =+0.01 dex. Thus, the stellar parameters can only be regarded as final if equations (2), (3) and (5) are satisfied and the model atmosphere is generated assuming a global metallicity of [m/h] = [Fe/H] NGC6752-mg9 + δa Fe i I = 1.65. While equations (2), (3) and (5) are primarily sensitive to T eff, ξ t and log g, respectively, in practice, all three equations are affected by small changes in any stellar parameter. Derivation of these strictly differential stellar parameters required multiple iterations (up to 20) where each iteration selected a single value for [m/h] and five values for each parameter, T eff,logg and ξ t,instepsof5k, 0.05 dex and 0.05 km s 1, respectively, i.e. 125 models per iteration. We then examined the output from the 125 models to see whether equations (2), (3) and (5) were simultaneously satisfied and whether the derived metallicity matched that of the model atmosphere. If not, the best model was identified and we repeated the process. If so, we conducted a final iteration in which we selected a single value for [m/h] and tested 11 values for each parameter, T eff,logg and ξ t, in steps of 1 K, 0.01 dex and 0.01 km s 1, respectively, i.e. 1331 models in the final iteration using a smaller step size for each parameter, and the best model was selected. As noted, this process was performed separately for the RGB tip sample and for the RGB bump sample. The strictly differential stellar parameters obtained using this pair of reference stars (RGB tip = NGC 6752-mg9, RGB bump = NGC 6752 11) are presented in Table 4. (We exclude the RGB tip star NGC 6752-mg1 because the stellar parameters did not converge. Specifically, the best solution required a value for log g beyond the boundary of the Castelli & Kurucz (2003) grid of model atmospheres.) Figs 2 and 3 provide examples of δa i,for Fe I and Fe II, versus lower excitation potential and reduced EW for the strictly differential stellar parameters for a representative RGB tip star and a representative RGB bump star, respectively. That is,

Precision abundance measurements in NGC 6752 3547 Table 4. Strictly differential stellar parameters and uncertainties when adopting the first set of reference stars (RGB tip = NGC 6752-mg9, RGB bump = NGC 6752 11). Name T eff σ log g σ ξ t σ [Fe/H] (K) (K) (cm s 2 ) (cms 2 ) (kms 1 ) (kms 1 ) (1) (2) (3) (4) (5) (6) (7) (8) NGC 6752-mg0 3919 20 0.16 0.01 2.24 0.05 1.69 NGC 6752-mg2 3938 22 0.23 0.01 2.13 0.05 1.67 NGC 6752-mg3 4066 19 0.53 0.01 1.93 0.04 1.65 NGC 6752-mg4 4081 18 0.54 0.01 1.90 0.04 1.65 NGC 6752-mg5 4100 17 0.56 0.01 1.93 0.04 1.66 NGC 6752-mg6 4151 19 0.65 0.01 1.88 0.04 1.63 NGC 6752-mg8 4284 14 0.93 0.01 1.73 0.04 1.65 NGC 6752-mg10 4291 12 0.92 0.01 1.70 0.03 1.66 NGC 6752-mg12 4315 13 0.96 0.01 1.76 0.04 1.66 NGC 6752-mg15 4339 13 1.01 0.01 1.76 0.04 1.66 NGC 6752-mg18 4380 15 1.07 0.01 1.71 0.04 1.66 NGC 6752-mg21 4437 13 1.16 0.01 1.69 0.05 1.65 NGC 6752-mg22 4444 14 1.19 0.01 1.71 0.04 1.64 NGC 6752-mg24 4505 17 1.30 0.01 1.72 0.07 1.68 NGC 6752-mg25 4471 15 1.24 0.01 1.74 0.07 1.69 NGC 6752 0 4706 12 1.85 0.01 1.44 0.02 1.65 NGC 6752 1 4719 11 1.94 0.01 1.37 0.02 1.65 NGC 6752 2 4739 12 1.95 0.01 1.35 0.02 1.66 NGC 6752 3 4749 13 2.00 0.01 1.34 0.02 1.73 NGC 6752 4 4794 13 2.08 0.01 1.37 0.02 1.66 NGC 6752 6 4795 11 2.10 0.01 1.32 0.02 1.64 NGC 6752 8 4930 15 2.29 0.01 1.31 0.03 1.67 NGC 6752 9 4795 21 2.09 0.01 1.40 0.04 1.73 NGC 6752 10 4811 10 2.11 0.01 1.35 0.02 1.67 NGC 6752 12 4822 13 2.15 0.01 1.34 0.02 1.68 NGC 6752 15 4830 12 2.23 0.01 1.34 0.02 1.65 NGC 6752 16 4875 15 2.24 0.01 1.31 0.03 1.66 NGC 6752 19 4892 12 2.32 0.01 1.31 0.02 1.71 NGC 6752 20 4899 12 2.32 0.01 1.30 0.02 1.65 NGC 6752 21 4884 14 2.32 0.01 1.30 0.03 1.69 NGC 6752 23 4912 12 2.33 0.01 1.25 0.02 1.67 NGC 6752 24 4911 17 2.39 0.01 1.14 0.03 1.74 NGC 6752 29 4923 13 2.40 0.01 1.30 0.02 1.71 NGC 6752 30 4919 12 2.47 0.01 1.24 0.02 1.66 Figure 2. Abundance differences, δa i, for the RGB tip star NGC 6752-mg8 (reference star NGC 6752-mg9) versus lower excitation potential (left) and reduced EW (right). Values for Fe I and Fe II are shown as black squares and red crosses, respectively. The blue dashed line in each panel is the linear least-squares fit to the data and we write the slope and associated uncertainty in each panel. In the right-hand panel, we also write Fe I Fe II = δa Fe i I δafe II i. these figures show the results when equations (2), (3) and (5) are simultaneously satisfied and the derived metallicity is the same as that used to generate the model atmosphere. In Figs 4 and 5 we compare the reference star stellar parameters (described in Section 2.3) and the strictly differential stellar parameters (described above) for the RGB tip and RGB bump samples, respectively, using the reference stars noted above. For the RGB tip sample, the average difference between the reference star and strictly differential values for T eff,logg, ξ t and [Fe/H] are very small; 7.53 ± 5.09 K, 0.015 ± 0.015 dex (cgs),

3548 D. Yong et al. Figure 3. Same as Fig. 2 but for the RGB bump star NGC 6752 15 (reference star NGC 6752 11). Figure 5. Same as Fig. 4 but for the RGB bump sample (reference star is NGC 6752 11). substantial change for any parameter, relative to the reference star stellar parameters.fort eff, the changes are within the uncertainties of the photometry. Figure 4. Differences in T eff,logg, ξ t and [Fe/H] between the reference star (old) and the strictly differential (new) stellar parameters for the RGB tip sample (reference star is NGC 6752-mg9). The mean difference is written in each panel. The green, magenta and blue colours represent populations a, b and c from Milone et al. (2013) (see Section 2.1 for details). 0.031 ± 0.004 km s 1 and 0.002 ± 0.004 dex, respectively. Comparably small differences in stellar parameters are obtained for the RGB bump sample. Therefore, an essential point we make here is that the strictly differential stellar parameters do not involve any 2.5 Chemical abundances Having obtained the strictly differential stellar parameters, we computed the abundances for the following species in every program star; Na, Si, Ca, Ti I, TiII, CrI, CrII, Ni,Y,La,NdandEu.Forthe elements La and Eu, we used spectrum synthesis and χ 2 analysis of the 5380 and 6645 Å lines, respectively, rather than an EW analysis since these lines are affected by hyperfine splitting (HFS) and/or isotope shifts. We treated these lines appropriately using the data from Kurucz & Bell (1995) and for Eu, we adopted the Lodders

Precision abundance measurements in NGC 6752 3549 Table 5. Differential abundances (Fe, Na, Si, Ca and Ti) when adopting the first set of reference stars (RGB tip = NGC 6752-mg9, RGB bump = NGC 6752 11). Star Fe σ Na σ Si σ Ca σ Ti I σ Ti II σ (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) NGC 6752-mg0 0.029 0.010 0.387 0.016 0.038 0.015 0.023 0.033 0.021 0.020 0.024 0.035 NGC 6752-mg2 0.011 0.011 0.014 0.008 0.039 0.010 0.021 0.049 0.050 0.024 0.036 0.039 NGC 6752-mg3 0.007 0.015 0.027 0.005 0.007 0.009 0.003 0.045 0.020 0.017 0.043 0.041 NGC 6752-mg4 0.010 0.014 0.041 0.010 0.030 0.010 0.008 0.038 0.023 0.012 0.043 0.045 NGC 6752-mg5 0.005 0.008 0.052 0.008 0.015 0.008 0.001 0.015 0.006 0.012 0.038 0.035 NGC 6752-mg6 0.032 0.009 0.123 0.002 0.042 0.009 0.049 0.040 0.052 0.011 0.095 0.065 NGC 6752-mg8 0.007 0.010 0.036 0.015 0.001 0.017 0.004 0.024 0.008 0.023 0.029 0.019 NGC 6752-mg10 0.007 0.010 0.013 0.004 0.004 0.007 0.005 0.017 0.023 0.010 0.027 0.033 NGC 6752-mg12 0.002 0.010 0.342 0.004 0.021 0.007 0.017 0.016 0.006 0.007 0.007 0.030 NGC 6752-mg15 0.001 0.009 0.044 0.009 0.008 0.010 0.008 0.011 0.009 0.007 0.009 0.023 NGC 6752-mg18 0.002 0.010 0.094 0.004 0.006 0.009 0.016 0.017 0.018 0.007 0.044 0.033 NGC 6752-mg21 0.018 0.009 0.282 0.009 0.043 0.009 0.032 0.013 0.012 0.009 0.057 0.031 NGC 6752-mg22 0.014 0.009 0.323 0.008 0.030 0.010 0.017 0.011 0.012 0.010 0.012 0.031 NGC 6752-mg24 0.023 0.016 0.345 0.035 0.049 0.009 0.040 0.012 0.034 0.009 0.047 0.059 NGC 6752-mg25 0.027 0.010 0.139 0.025 0.008 0.010 0.026 0.023 0.045 0.009 0.023 0.039 NGC 6752 0 0.030 0.010 0.335 0.033 0.096 0.019 0.050 0.010 0.023 0.011 0.052 0.012 NGC 6752 1 0.025 0.009 0.366 0.020 0.008 0.013 0.031 0.010 0.003 0.011 0.034 0.012 NGC 6752 2 0.020 0.008 0.384 0.015 0.055 0.012 0.038 0.008 0.001 0.008 0.031 0.014 NGC 6752 3 0.049 0.012 0.444 0.016 0.044 0.007 0.044 0.009 0.052 0.013 0.036 0.017 NGC 6752 4 0.017 0.015 0.352 0.021 0.026 0.021 0.065 0.011 0.007 0.013 0.034 0.017 NGC 6752 6 0.036 0.014 0.262 0.017 0.032 0.008 0.060 0.011 0.027 0.013 0.042 0.014 NGC 6752 8 0.010 0.014 0.323 0.012 0.045 0.017 0.027 0.010 0.030 0.012 0.018 0.013 NGC 6752 9 0.048 0.025 0.396 0.056 0.049 0.011 0.038 0.013 0.062 0.016 0.045 0.018 NGC 6752 10 0.013 0.011 0.357 0.020 0.016 0.012 0.039 0.014 0.007 0.019 0.032 0.014 NGC 6752 12 0.000 0.013 0.065 0.009 0.012 0.016 0.003 0.010 0.023 0.013 0.027 0.016 NGC 6752 15 0.033 0.012 0.355 0.075 0.002 0.012 0.022 0.011 0.006 0.015 0.042 0.015 NGC 6752 16 0.021 0.016 0.091 0.014 0.005 0.018 0.008 0.011 0.001 0.015 0.007 0.016 NGC 6752 19 0.029 0.012 0.190 0.008 0.048 0.010 0.029 0.008 0.046 0.011 0.024 0.012 NGC 6752 20 0.029 0.012 0.454 0.015 0.031 0.015 0.051 0.009 0.020 0.013 0.037 0.012 NGC 6752 21 0.007 0.013 0.063 0.003 0.019 0.018 0.010 0.011 0.010 0.014 0.011 0.013 NGC 6752 23 0.016 0.012 0.272 0.019 0.032 0.012 0.033 0.009 0.002 0.013 0.024 0.015 NGC 6752 24 0.058 0.016 0.408 0.010 0.107 0.020 0.048 0.015 0.078 0.011 0.081 0.018 NGC 6752 29 0.026 0.012 0.421 0.032 0.101 0.020 0.025 0.009 0.064 0.021 0.043 0.012 NGC 6752 30 0.025 0.011 0.161 0.010 0.007 0.013 0.056 0.012 0.003 0.015 0.051 0.014 Notes. In order to place the above values on to an absolute scale, the absolute abundances we obtain for the reference stars are given below. We caution, however, that the absolute scale has not been critically evaluated (see Section 2.5 for more details). NGC 6752-mg9: A(Fe) = 5.85, A(Na) = 4.86, A(Si) = 6.23, A(Ca) = 4.99, A(Ti I) = 3.54, A(Ti II) = 3.59. NGC 6752 11: A(Fe) = 5.84, A(Na) = 4.84, A(Si) = 6.24, A(Ca) = 4.97, A(Ti I) = 3.50, A(Ti II) = 3.72. (2003) solar isotope ratios. The log gf values for the La and Eu lines were taken from Lawler, Bonvallet & Sneden (2001a) and Lawler et al. (2001b), respectively. We used equation (1) to obtain the abundance difference (between the program star and the reference star) for any line. For a particular species, X, the average abundance difference is δa X i which we write as X, i.e. as defined in equation (4) above. In Tables 5 and 6, we present the abundance differences for each element in all program stars. In order to put these abundance differences on to an absolute scale, in these tables we also provide the A(X) abundances for the reference stars when using the stellar parameters in Table 1. The new [X/Fe] values are in very good agreement with our previously published values (Grundahl et al. 2002; Yong et al. 2003, 2005), although we have not attempted to reconcile the two sets of abundances. For Na, the range in abundance is 0.90 dex, in good agreement with our previously published values. We did not attempt to remeasure the abundances of other light elements, O, Mg and Al, as multiple lines could not be measured in all stars. Additionally, given the well-established correlations between the abundances of these elements, we believe that Na provides a reliable picture of the light element abundance variations in this cluster. The interested reader can find our abundances for N, O, Mg and Al in Grundahl et al. (2002) and Yong et al. (2003, 2008). (C measurements in the RGB bump sample are ongoing and will be presented in a future work.) As mentioned, Meléndez et al. (2012) considered the relative ionization equilibria for Ti and Cr when establishing the strictly differential stellar parameters. Having measured the Ti and Cr abundances from neutral and ionized lines, we are now in a position to examine Ti I Ti II = δa Ti i I Ti II δai and Cr I Cr II = δa Cr i I Cr II δai. In Fig. 6, we plot Ti I Ti II and Cr I Cr II versus log g for both samples of stars. In this figure, it is clear that ionization equilibrium is not obtained for Ti or Cr and that there are trends between Ti I Ti II versus log g and Cr I Cr II versus log g. Nevertheless, we are satisfied with our approach which used only Fe lines to establish the differential stellar parameters. We expect that inclusion of Ti and Cr ionization equilibrium would have resulted in very small adjustments to the stellar parameters and to the differential chemical abundances. Finally, as it will be shown later,

3550 D. Yong et al. Table 6. Differential abundances (Cr, Ni, Y, La, Nd and Eu) when adopting the first set of reference stars (RGB tip = NGC 6752-mg9, RGB bump = NGC 6752 11). Star Cr I σ Cr II σ Ni σ Y σ La σ Nd σ Eu σ (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13) (14) (15) NGC 6752-mg0 0.013 0.059 0.018 0.077 0.030 0.023 0.022 0.037 0.028 0.013 0.011 0.042 0.002 0.012 NGC 6752-mg2 0.053 0.087 0.068 0.074 0.000 0.021 0.087 0.045 0.081 0.017 0.051 0.058 0.012 0.013 NGC 6752-mg3 0.042 0.046 0.042 0.035 0.005 0.023 0.074 0.036 0.106 0.016 0.046 0.061 0.063 0.013 NGC 6752-mg4 0.050 0.046 0.055 0.033 0.013 0.019 0.075 0.024 0.073 0.015 0.058 0.042 0.056 0.014 NGC 6752-mg5 0.034 0.037 0.023 0.029 0.001 0.011 0.006 0.035 0.140 0.016 0.029 0.026 0.027 0.014 NGC 6752-mg6 0.028 0.042 0.044 0.024 0.038 0.023 0.098 0.028 0.109 0.017 0.067 0.053 0.060 0.014 NGC 6752-mg8 0.029 0.035 0.095 0.085 0.007 0.014 0.015 0.013 0.087 0.016 0.026 0.016 0.053 0.016 NGC 6752-mg10 0.009 0.022 0.055 0.074 0.001 0.012 0.079 0.020 0.020 0.017 0.019 0.025 0.032 0.016 NGC 6752-mg12 0.005 0.013 0.014 0.006 0.003 0.008 0.006 0.020 0.036 0.016 0.000 0.021 0.013 0.016 NGC 6752-mg15 0.027 0.011 0.019 0.014 0.006 0.007 0.001 0.004 0.042 0.016 0.015 0.013 0.013 0.014 NGC 6752-mg18 0.026 0.016 0.032 0.014 0.007 0.010 0.014 0.026 0.005 0.018 0.011 0.028 0.007 0.017 NGC 6752-mg21 0.003 0.023 0.021 0.012 0.002 0.008 0.068 0.023 0.059 0.017 0.010 0.022 0.037 0.017 NGC 6752-mg22 0.017 0.042 0.007 0.039 0.009 0.009 0.047 0.018 0.049 0.017 0.013 0.016 0.008 0.018 NGC 6752-mg24 0.033 0.013 0.060 0.013 0.024 0.008 0.062 0.015 0.005 0.016 0.032 0.018 0.018 0.018 NGC 6752-mg25 0.023 0.021 0.046 0.014 0.043 0.010 0.038 0.018 0.108 0.015 0.051 0.026 0.003 0.018 NGC 6752 0 0.058 0.012 0.112 0.053 0.020 0.009 0.044 0.018 0.018 0.012 0.018 0.015 0.123 0.024 NGC 6752 1 0.037 0.014 0.077 0.060 0.010 0.014 0.026 0.027 0.060 0.012 0.009 0.025 0.068 0.026 NGC 6752 2 0.009 0.012 0.038 0.005 0.003 0.008 0.017 0.023 0.032 0.011 0.009 0.029 0.180 0.023 NGC 6752 3 0.053 0.023 0.053 0.029 0.057 0.013 0.143 0.009 0.039 0.012 0.110 0.025 0.089 0.025 NGC 6752 4 0.014 0.023 0.062 0.046 0.003 0.012 0.018 0.022 0.009 0.010 0.014 0.027 0.328 0.025 NGC 6752 6 0.038 0.027 0.068 0.052 0.004 0.012 0.005 0.025 0.027 0.013 0.041 0.035 0.208 0.025 NGC 6752 8 0.019 0.016 0.061 0.055 0.004 0.008 0.026 0.026 0.064 0.010 0.033 0.014 0.179 0.029 NGC 6752 9 0.039 0.026 0.028 0.044 0.054 0.016 0.089 0.012 0.014 0.011 0.064 0.023 0.149 0.025 NGC 6752 10 0.029 0.022 0.016 0.022 0.016 0.014 0.016 0.013 0.076 0.012 0.013 0.025 0.185 0.029 NGC 6752 12 0.004 0.021 0.075 0.065 0.016 0.010 0.097 0.021 0.006 0.011 0.020 0.032 0.008 0.028 NGC 6752 15 0.024 0.021 0.070 0.021 0.005 0.013 0.046 0.026 0.005 0.011 0.010 0.025 0.082 0.034 NGC 6752 16 0.016 0.019 0.012 0.024 0.007 0.013 0.048 0.015 0.031 0.013 0.045 0.031 0.001 0.039 NGC 6752 19 0.036 0.021 0.016 0.048 0.052 0.010 0.107 0.013 0.018 0.011 0.049 0.024 0.004 0.042 NGC 6752 20 0.024 0.018 0.038 0.019 0.007 0.007 0.012 0.014 0.054 0.012 0.011 0.026 0.057 0.042 NGC 6752 21 0.014 0.018 0.052 0.025 0.032 0.009 0.013 0.015 0.087 0.011 0.023 0.019 0.032 0.039 NGC 6752 23 0.006 0.025 0.102 0.036 0.026 0.010 0.016 0.010 0.028 0.011 0.004 0.011 0.033 0.040 NGC 6752 24 0.056 0.019 0.031 0.020 0.089 0.010 0.135 0.018 0.050 0.012 0.075 0.016 0.141 0.050 NGC 6752 29 0.036 0.020 0.051 0.042 0.056 0.011 0.082 0.022 0.094 0.012 0.054 0.021 0.062 0.033 NGC 6752 30 0.029 0.016 0.048 0.037 0.007 0.010 0.000 0.032 0.047 0.011 0.025 0.017 0.235 0.031 Notes. In order to place the above values on to an absolute scale, the absolute abundances we obtain for the reference stars are given below. We caution, however, that the absolute scale has not been critically evaluated (see Section 2.5 for more details). NGC 6752-mg9: A(Cr I) = 3.99, A(Cr II) = 4.10, A(Ni) = 4.56, A(Y) = 0.67, A(La) = 0.39, A(Nd) = 0.06, A(Eu) = 0.75. NGC 6752 11: A(Cr I) = 3.84, A(Cr II) = 4.12, A(Ni) = 4.54, A(Y) = 0.66, A(La) = 0.29, A(Nd) = 0.06, A(Eu) = 0.80. Ti and Cr have considerably higher uncertainties such that it may be better to rely only upon Fe for ionization balance. 2.6 Error analysis To determine the errors in the stellar parameters, we adopted the following approach. For T eff, we determined the formal uncertainty in the slope between δa Fe i I and the lower excitation potential. We then adjusted T eff until the formal slope matched the error. The difference between the new T eff and the original value is σ T eff.for the RGB tip and RGB bump stars, the average values of σ T eff were 7.53 and 21.74 K, respectively. For log g, we added the standard error of the mean for Fe I and Fe II in quadrature and then adjusted log g until the quantity Fe I Fe II, from equation (5), was equal to this value. The difference between the new log g and the original value is σ log g. For the RGB tip and RGB bump stars, the average values of σ log g were 0.015 and 0.009 dex, respectively. For ξ t,we measured the formal uncertainty in the slope between δa Fe i I and the reduced EW. We adjusted ξ t until the formal slope was equal to this value. The difference between the new and old values is σξ t. Average values for σξ t for the RGB tip and RGB bump samples were 0.031 and 0.018 km s 1, respectively. Uncertainties in the element abundance measurements were obtained following the formalism given in Johnson (2002), which we repeat here for convenience, and we note that this approach is very similar to that of McWilliam et al. (1995) and Barklem et al. (2005). ( logɛ σlogɛ 2 = σ rand 2 + T + 2 [ ( logɛ T ) 2 ( ) logɛ 2 ( ) logɛ 2 σt 2 + σlogg 2 logg + σξ 2 ξ )( ) logɛ σ T logg + logg ( logɛ ξ )( ) logɛ σ loggξ logg ( )( ) ] logɛ logɛ + σ ξt. (6) ξ T The covariance terms, σ T logg, σ loggξ and σ ξt, were computed using the approach of Johnson (2002). These abundance uncertainties are included in Tables 5, 6, 8 and 9. For La and Eu, the abundances were obtained from a single line. For these lines, we adopt the

Precision abundance measurements in NGC 6752 3551 Figure 6. Ti I Ti II (upper panels) and Cr I Cr II (lower panels) for the RGB tip star sample (left-hand panels) and the RGB bump star sample (right-hand panels). (These results are obtained when using the reference stars RGB tip = NGC 6752-mg9 and RGB bump = NGC 6752 11.) The colours are the same as in Fig. 4. 1σ fitting error from the χ 2 analysis in place of the random error term, σ rand (standard error of the mean). We note that these formal uncertainties, which take into account all covariance error terms, are below 0.02 dex for many elements in many stars, reaching values as low as 0.01 dex for a number of elements including Si, Ti I,Ni and Fe. Note that in Fig. 6, we regard Ti I Ti II as an abundance ratio between Ti I and Ti II, and thus, we compute the error terms according to the relevant equations in Johnson (2002) which we again repeat here for convenience. σ 2 (A/B) = σ 2 (A) + σ 2 (B) 2σ A,B. (7) The covariance between two abundances is given by ( )( ) ( logɛa logɛb σ A,B = σt 2 T T + logɛa logg ( )( logɛa logɛb + ξ ξ + ( logɛa logg ) σ 2 ξ + [ ( logɛa T )( logɛb logg )( ) ] [ ( ) logɛb logɛa σ T logg + T ξ ) σ 2 logg )( ) logɛb logg ( ) ( )( ) ] logɛb logɛa logɛb + σ ξlogg. (8) logg logg ξ 3 RESULTS AND DISCUSSION 3.1 Trends versus T eff In Figs 7 9, we plot Fe Cr II, and Ni versus T eff, respectively. In these figures, the RGB tip sample and the RGB bump sample are in the upper and lower panels, respectively. In each panel, we show the mean and the abundance dispersion for X (σ A in these figures). We also determine the linear least-squares fit to the data and write the slope, uncertainty and abundance dispersion about the fit (σ B in these figures). For the subset of RGB tip stars within 100 and 200 K of the reference star, we compute and write the mean abundance and abundance dispersions (σ A and σ B ). Similarly, for Figure 7. Fe versus T eff for the RGB tip star sample (upper panel) and the RGB bump star sample (lower panel). In both panels, we show the location of the reference star as a black cross. We write the mean abundance and standard deviation (σ A ) for stars within 100 and 200 K of the reference star as well as for the full sample. The red dashed line is the linear least-squares fit to the data. The slope, uncertainty and dispersion (σ B ) about the linear fit are written. We also write the average abundance error, σ Fe, for each sample. (These results are obtained when using the reference stars RGB tip = NGC 6752-mg9 and RGB bump = NGC 6752 11.) The colours are the same as in Fig. 4. the subset of RGB bump stars within 50 and 100 K of the reference star, we write the same quantities. Finally, we also write the average abundance error, σ X, for a particular element for the RGB tip and RGB bump samples. Fe and Ni (Figs 7 and 9) are examples where the average abundance errors are very small, 0.01 dex. Cr II (Fig. 8) is the element that shows the highest average abundance error, 0.04 dex. Rather than showing similar figures for every element, in Fig. 10 we plot (i) the average abundance error ( σ X ), (ii) the abundance dispersion (σ A ) and (iii) the abundance dispersion about the linear fit to X versus T eff (σ B ), for all elements in the RGB tip sample (upper) and the RGB bump sample (lower). The main point to take from this figure is that we have achieved very high precision chemical abundance measurements from our strictly differential analysis for this sample of giant stars in the globular cluster NGC 6752. For the RGB tip sample, the lowest average abundance error is for Fe ( σ Fe =0.011 dex) and the highest value is for Cr II ( σ Cr II =0.052 dex). For the RGB bump sample the lowest average abundance errors are for Fe and La ( σ Fe, La =0.013 dex) while the highest value is for Cr II ( σ Cr II =0.041 dex). Another aspect to note in Fig. 10 is that the measured dispersions (σ A and σ B ) for many elements appear to be considerably larger than the average abundance error. We interpret such a result as evidence for a genuine abundance dispersion in this cluster, although another possible explanation is that we have systematically underestimated the errors.

3552 D. Yong et al. Figure 10. Average abundance errors ( σ X, filled black circles), abundance dispersions (σ A, red crosses) and abundance dispersions about the linear fits as seen in Figs 7 9 (σ B, blue triangles) for all species in the RGB tip sample (upper panel) and RGB bump sample (lower panel). (These results are obtained when using the reference stars RGB tip = NGC 6752-mg9 and RGB bump = NGC 6752 11.) Figure 8. Same as Fig. 7 but for Cr II versus T eff. 3.2 X versus Na Figure 9. Same as Fig. 7 but for Ni versus T eff. In Figs 11 and 12, we plot Fe versus Na and Si versus Na,respectively. In both figures, the RGB tip sample and the RGB bump Figure 11. Fe versus Na for the RGB tip star sample (upper) and the RGB bump star sample (lower). The red dashed line is the linear leastsquares fit to the data (slope and error are written). We write the dispersion in the y-direction (σ A ), the dispersion about the linear fit (σ B )andthe average abundance error, σ Fe, for each sample. (These results are obtained when using the reference stars RGB tip = NGC 6752-mg9 and RGB bump = NGC 6752 11.) The colours are the same as in Fig. 4. sample are found in the upper and lower panels, respectively. (Here one readily sees that the populations a (green), b (magenta) and c (blue) identified by Milone et al. (2013) from colour magnitude diagrams have distinct Na abundances.) We measure the linear

Precision abundance measurements in NGC 6752 3553 Figure 13. Slope of the fit to X versus Na,forX= Si to Eu, for the RGB tip sample (upper) and the RGB bump sample (lower). The colours represent the significance of the slope, i.e. the magnitude of the gradient divided by the uncertainty. (These results are obtained when using the reference stars RGB tip = NGC 6752-mg9 and RGB bump = NGC 6752 11.) Figure 12. Same as Fig. 11 but for Si versus Na. least-squares fit to the data and in each panel we write (i) the slope and uncertainty, (ii) the abundance dispersion (σ A ), (iii) the abundance dispersion about the linear fit to X versus Na (σ B ) and (iv) the average abundance error ( σ X ). Consideration of the slope and uncertainty of the linear fits reveals that while the amplitude may be small, there are statistically significant correlations between Fe and Na for the RGB bump sample and between Si and Na for the RGB tip and RGB bump samples. The results for Si confirm and expand on the correlations found between Si and Al (Yong et al. 2005) and between Si and N (Yong et al. 2008). In Fig. 13, we plot the slope of the linear fit to X versus Na for all elements in the RGB tip sample (upper) and the RGB bump sample (lower). With the exception of La and Eu (for the RGB tip sample), all the gradients are positive. For La and Eu in the RGB tip sample, the negative gradients are not statistically significant, <1σ. Assuming an equal likelihood of obtaining a positive or negative gradient, the probability of obtaining 22 positive values in a sample of 24 is 10 5. Based on the slope and uncertainty, we obtain the significance of the correlations; 8 of the 24 elements exhibit correlations that are significant at the 5σ level or higher. 4 Therefore, the first main conclusion we draw is that there are an unusually large number of elements that show positive correlations for X versus Na, and that an unusually large fraction of these correlations are of high statistical significance. We interpret this result as further evidence for a genuine abundance dispersion in this cluster. On this occasion, it is highly unlikely that such correlations could arise 4 We also performed linear fits to these data using the GAUSSFIT program for robust estimation (Jefferys, Fitzpatrick & McArthur 1988). While we again find positive gradients for 22 of the 24 elements, on average the significance of these correlations decreases from 3.9σ (least-squares fitting) to 2.6σ (robust fitting). When using the GAUSSFIT robust fitting routines, 3 of the 24 elements exhibit correlations that are significant at the 5σ level or higher. from underestimating the errors. NLTE corrections for Na, using improved atomic data, have been published by Lind et al. (2011b). The corrections are negative and strongly dependent on line strength; for a given T eff :log g:[fe/h], stronger lines have larger amplitude (negative) NLTE corrections. Had we included these corrections, the X versus Na(NLTE) gradients would be even steeper. We also note that the gradients are, on average, of larger amplitude and of higher statistical significance for the RGB bump sample compared to the RGB tip sample. Other than spanning a different range in stellar parameters, one notable difference between the two samples is that the RGB bump sample exhibits a larger range in Na than does the RGB tip sample. In particular, the numbers of RGB tip and RGB bump stars with Na 0.20 dex are 5 and 14, respectively. (Equivalently, the numbers of stars in the Milone et al. (2013) b and c populations are considerably larger in the RGB bump sample compared to the RGB tip sample.) Thus, we speculate that the RGB bump stars are the more reliable sample (based on the sample size and abundance distribution) from which to infer the presence of any trend between X and Na. We conducted the following test in order to check whether differences in gradients for X versus Na between the RGB tip and RGB bump samples can be attributed to differences in the Na distributions between the two samples. We start by assuming that the RGB bump sample provides the correct slope. For a given element, we consider the gradient and uncertainty for X versus Na and draw a random number from a normal distribution (centred at zero) whose width corresponds to the uncertainty. We add that random number to the gradient to obtain a new RGB bump gradient for X versus Na. For each RGB tip star, we infer the corresponding X using this new RGB bump gradient. We then draw another random number from a normal distribution (centred at zero) of width corresponding to the measurement uncertainty, σ X,and add that number to the X value inferred. For a given element, we measure the gradient and uncertainty for this new set of X values. We repeated the process for 1000 000 realizations. Our expectation is that these Monte Carlo simulations predict the gradient for X